46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference
18 - 21 April 2005, Austin, Texas
AIAA 2005-1870
AIAA 2005-1870
Bi-stable Cylindrical Space Frames
H. Ye and S. Pellegrino
University of Cambridge, Cambridge, CB2 1PZ, UK
46th AIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics, and
Materials Conference
18-21 April 2005
Austin, Texas
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344
Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Bi-stable Cylindrical Space Frames
H. Ye∗ and S. Pellegrino†
University of Cambridge, Cambridge, CB2 1PZ, UK
This paper presents a novel kind of bi-stable structure: bi-stable space frames, based
on a double-layer cylindrical architecture. An analytical method to carry out a preliminary assessment of the bi-stability of such space frames is introduced here and a series
of bi-stable space frames found by this method are presented. Non-linear finite element
simulations have been used to test one of the space frames and confirm the analytical
predictions.
Introduction
Several bi-stable structures, which have two discrete
stable configurations, have recently been found and
the application of these structures to next-generation
deployable and adaptive structures is currently being
investigated at Cambridge. Many of the bi-stable
structures that are known so far have the form of
a metallic or composite cylindrical shells [1] [2] and
assemblies of hinged bars and tape springs that are
bi-stable are also known [3].
This paper will present a new kind of bi-stable
structure –a bi-stable space frame– based on a doublelayer cylindrical architecture. A method to carry out
a preliminary assessment of the bi-stability of this
kind of structure will be introduced and explained.
A series of space frames having bi-stable properties
have been found.
Several key issues concerning
the design of this kind of bi-stable structure have
been revealed. Based on the geometrical estimation
method, a cylindrical space frame that is potentially
bi-stable has been selected and a series of detailed
geometrically non-linear finite element simulations
of this structure, varying several design parameters,
have been done to investigated its bi-stability. Finally,
a couple of physical models built by rapid prototyping
technology will be presented.
we introduce the equivalent continuum modulus matrix for some simple examples.
Consider the two-dimensional lattice structure
shown in Figure 1(a). It is assumed that all squares
forming this structure are identical, and have the equal
side length L and axial stiffness AE. A Cartesian coordinate system has been defined for the structure,
whose axes are parallel to the sides of the space frame.
As in continuum mechanics, we can introduce the
equivalent modulus matrix D to describe its in-plane
structural properties. The relationship between the
in-plane axial and shearing force per unit length and
the corresponding deformation variables is given by




Nx
²x
 Nxy  = D  ²xy 
(1)
Ny
²y
in which the matrix on the right of the equation is the
equivalent modulus matrix:


1 0 0
AE 
0 0 0 
D=
(2)
L
0 0 1
Ny
Ny
Nxy
Nxy
Nxy
1
L
Nxy
1
L
Nx
Nx
Simple Geometrical Approach
Structural properties of space frames
A space frame is a structure system whose overall
shape is two- or three-dimensional. It consists of linear elements that carry loads in a three dimensional
way and these elements form a series of repeating and
regular units [4]. For any space frame, the general
structural properties can be usefully described in term
of an equivalent continuum, which simulates the properties of basic repeating units in the space frame. Here
∗ Research
Student.
of Structural Engineering, Department of Engineering, Trumpington Street. Associate Fellow AIAA.
† Professor
Copyright °
c 2005 by H. YE and S. Pellegrino. Published by
the American Institute of Aeronautics and Astronautics, Inc. with
permission.
1
1
(a)
(b)
Fig. 1 Two dimensional repeating lattice based on
(a) square and (b) right-angle triangle.
Similarly, for the two dimensional space frame based
on right-angle triangles shown in Figure 1(b), the
equivalent modulus matrix D can also be obtained by
superposing the contribution of the additional diagonal bar to the modulus matrix in Equation (2):


1
1
1
√
√
1 + 2√
2
2 2
2 2
AE 

1
1
1
√
√
(3)
D=
 2√2

2
2
2
2
L
1
1
1
√
√
√
1
+
2 2
2 2
2 2
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American Institute of Aeronautics and Astronautics Paper 2005-1870
Modulus matrices for lattices having several other inplane forms have also also been derived [5].
KXY
(Kx0,Kxy0)
y0
My
Mx
(Kc0,0)
x0
θ0
(Ky0,-Kxy0)
Mx
X
(a)
d
KXY
y0
y
(Kx0,Kxy0)
My
(Kx,Kxy)
Fig. 2 Three dimensional space frame with identical top and bottom layer
Y
For a three dimensional double-layer space frame as
Figure 2 shows— in which the two layers are identical
to the two-dimensional lattice considered previously
(Figure 1(b)), but are now separated by a distance
d, we can also develop an equivalent modulus matrix
to describe its structural behavior under pure outof-plane bending and twisting. If plane sections are
assumed to remain plane and bars connecting the top
and bottom layers provides a shear-rigid connection
between layers, the relationship between bending and
twisting moment per unit length and the corresponding curvatures and twist in the structure can be related
as:




Mx
κx
 Mxy  = D∗  κxy 
(4)
My
κy
where the modulus matrix D∗ can be given by modifying the modulus matrix of the corresponding lattice
in Equation (2):
d2
D
2
(5)
It can be shown that Equation (5) is valid for any
double layer space frame with equal top and bottom
layers. Equations (2) and (5) show that the elastic
structural properties of any double-layer space frame
can simply be derived from their corresponding two
dimensional architecture.
Deformation of cylindrical space frame
We now consider the main object of our
investigation—a double-layer space frame with cylindrical shape, whose mid-plane lies on the surface of
a cylinder. To investigate its bi-stability, it is necessary to know the strain energy of the structure in all
deformed configurations.
For simplicity, we initially assume that only a uniform and inextensional deformation of this space frame
is allowed, and hence the mid-plane of the space frame
has to remain on a cylindrical surface at all stages, as
shown in Figure 3. A fixed global coordinate system
X − Y has been set in Figure 3 while a local coordinate system x − y attached to the space frame will be
allowed to rotate along with the structure itself. The
initial configuration of the space frame is represented
x
θ
1/Kc
(0,0)
2θ
KX,KY
2θ0
(Kc,0)
x0
X
D∗ =
KX,KY
2θ0
(0,0)
1/Kc0
Y
(Ky,-Kxy)
(Ky0,-Kxy0)
(b)
Fig. 3 Inextensional deformation of a cylindrical
space frame: (a)Initial configuration (b) Deformed
configuration)
by the curvature of the initial cylinder κc0 and the initial rotating angle θ0 . As shown in Figure 3(b), the
deformed configuration will be denoted by curvature
of current cylinder κc and the rotated angle θ.
The strain energy associated with any amount of
bending and twisting of this structure (but note that
the structure is not allowed to bend simultaneously in
two different directions, because of the assumed inextensional deformation) is expressed by the equation



∆κx
1 0 0
1
U = [∆κx ∆κxy ∆κy ] D∗  0 2 0   ∆κxy 
2
0 0 1
∆κx
(6)
where D∗ is the equivalent modulus matrix of the
space frame. ∆κx ,∆κxy and ∆κy are the changes of
curvature of the structure in the local coordinate system.
∆κx
∆κxy
∆κy
= κx − κx0
= κxy − κxy0
= κx − κy 0
(7)
(8)
(9)
These curvatures can be obtained by Mohr’s circle
(Figure 3):
κc0
(1 + cos(2θ0 ))
2
κc0
= −
sin(2θ0 )
2
κc0
=
(1 − cos(2θ0 ))
2
κx0
=
κxy0
κy 0
(10)
(11)
(12)
and
κx
κxy
κy
κc
(1 + cos 2(θ0 + θ))
2
κc
= − sin 2(θ0 + θ)
2
κc
=
(1 − cos 2(θ0 + θ))
2
=
(13)
(14)
(15)
For any given cylindrical space frame, with specified initial configuration κc0 and θ0 , we can consider
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American Institute of Aeronautics and Astronautics Paper 2005-1870
all possible inextensionally deformed configurations by
varying the curvature of the cylinder κc and the twisting angle θ. Therefore, the strain energy of all configurations can be calculated and their stability can be
examined from the contour plot of the energy.
5
180
13 Second
9
11
5
13
7
θ (deg)
0.2
3
7
5
60
11
13
11
7
5
5
7
1
2
1
3
4
5
6
3
1
0.2
8
7
13
7 9
.
3
0
9
11
Initial stable configuration
9
9
20
5
25
13
40
1
13
9
11
9
5
3
1
3
7
5
9
10
κ (m−1)
c
Fig. 4
Contour plot of strain energy of a space
frame with lattice corresponding to Figure 1(a).
κc0 = 8.3 m−1
θ0 = 45◦
L = 8 mm
d = 6 mm
A = 0.196 mm2
E = 2.5 GPa
180
0.2
6
11
160
2
4
13
6
9
8
9
140
40
30
11
80
30
.
25
13
9
8
θ (deg)
25
35
9
100
9
60
35
40
11
1
2
3
98
.
2
6
0
5
13
6
4
11
13
20
4
40
Initial stable configuration
2
3
9
8
5
6
0.2
8
7
0
1113
9
6 8
4
2
9
10
−1
κc (m )
Fig. 5
Contour plot of strain energy of a space
frame with lattice corresponding to Figure 1(b)
of diagonal bars (as shown in Figure 1) may have crucial effects on determining the second configuration.
The energy analysis method has been applied on a series of space frames to investigate the effects of the
axial stiffness of diagonal bars. The results are summarized in Table 2. Here the axial stiffness of diagonal
bars is denoted as AE1 while those of bars on the sides
of squares are denoted as AE2 . The results clearly
show that the second configuration is determined by
the ratio of axial stiffness. With the increasing of the
ratio, the curvature of second configuration increases
up to the same value as the initial configuration. The
rotating angle θ, however, remains the same.
Table 2
Design rules for bi-stable space frames
The fact that the above two space frames with different in-plane architectures have two varied second
stable configurations indicates that the axial stiffness
11
Second stable configuration
120
11
the strain energy analysis method mentioned above
has been performed to investigate the bi-stability. The
contour plots of the strain energy of each space frame
under inextensional deformation are shown in Figure
4 and Figure 5 respectively. It is clearly shown that
there are two local minimum energy points in each
diagram, which correspond to two stable configurations of each structure. Figure 4 shows the energy
of a double-layer space frame with the same in-plane
architecture as shown in Figure 1(a). One local minimum strain energy point is at κc0 = 8.3 m−1 and
θ0 = 0, corresponding to the initial stress-free configuration. The other local minimum energy point is at
κc1 = 8.3 m−1 and θ = 90◦ . By referring to Figure 3,
it is found that the mid-plane of the second configuration lies on the same cylinder as the initial one, but
rotated by 90◦ in the X − Y plane. Figure 5 shows the
energy of a double-layer space frame with the same
in-plane architecture in Figure 1(b). One local minimum strain energy point in this figure is also at initial
configuration κc0 = 8.3 m−1 and θ0 = 0. The second
local minimum energy point is at κc1 = 3.7 m−1 and
θ = 90◦ . Here, the mid-plane of the second stable
configuration lies on a cylinder different to the initial state. It is also observed that in Figure 4, both
local minimum energy points have the same energy
value. This space frame is therefore a symmetrically
bi-stable structure, meaning a structure that has two
stable states with the same energy level. In Figure 5,
the two stable configurations have two different energy
levels. This space frame is therefore a asymmetrically
bi-stable structure [6].
8
13
11
2
6
9 8
13
4
13
Initial curvature of mid-plane
Initial rotating angle
Side length of square
Depth of space frame
Cross section area (all bars)
Young’s modulus of material
.
1
13
9
7
5
3
80
25
stable configuration
11
7
9
We choose two example cases here to demonstrate
this method of energy analysis. Two cylindrical
double-layer space frames with the in-plane two dimensional architectures shown in Figure 1(a) and (b)
have been selected and their structural properties are
set as shown in Table 1. For these two space frames,
Structural properties of space frames
11
13
7
11
120
5
7
911
13
9
9
Two examples of bi-stable space frame
1
3
5
140
100
Table 1
0.2
1
3
7
160
Effects of axial stiffness
AE2 /AE1
0.5
1
2
3
5
10
∞
κc0 (m−1 )
θ0 (deg)
8.3
45
8.3
45
8.3
45
8.3
45
8.3
45
8.3
45
8.3
45
κc1 (m−1 )
θ(deg)
2.7
90
3.7
90
4.7
90
5.5
90
6.3
90
7.2
90
8.3
90
Further research on additional space frame with
other in-plane architectures has also been carried out.
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American Institute of Aeronautics and Astronautics Paper 2005-1870
The parameters of the models are set to the same as in
Table 1. The analysis results show that for the space
frames with the first two in-plane architectures (Figures 6(a) and (b)), the second stable configuration remains the same at κc1 = 8.3 m−1 and θ = 90◦ . But for
a model having the architecture shown in Figure 6(c),
κc1 = 5.4 m−1 and θ = 79◦ . For the model with architecture shown in Figure 6(d), κc1 = 5.2 m−1 and
θ = 94◦ . Combining the above results, it is demonstrated that the second configuration of this kind of
bi-stable spaces frame is determined by the axial stiffness and the orientation of the diagonal bars. In other
words, it is the diagonal bars that decide the shape of
the second configuration.
the parameters specified as the same as those in the
simple geometrical calculation presented in Table 1.
Ny
Ny
Nxy
1
Fig. 7
frame
Nxy
Nxy
Nxy
1
2L
L
Nx
L
2L
Simulation techniques
1
1
(a)
(b)
The coordinates of the nodes in the structure for
each of the previously obtained configurations have
been calculated. In the FE analysis, the structure
has been switched from the initial configuration to the
second one by applying temporary displacement constraints on 5 nodes in the structure. After the second
configuration had been approached sufficiently closely,
all temporary constraints are removed, and only six
rigid body constraints are left, in order to test the stability of the structure in this new configuration. All the
analysis is carried out as nonlinear geometrical simulations in ABAQUS.
Ny
Ny
Nxy
Nxy
Nxy
Nxy
1
2L
L
L
2L
Single units of a three-dimensional space
1
1
(c)
(d)
Nx
Fig. 6
Analysis results and discussion
Finite Element Modelling
Based on a series of assumptions, the above geometrical approach gives a preliminary assessment of the
bi-stability of cylindrical space frames. A more realistic finite element model has been established here to
confirm and test the bi-stability of space frame. The
analysis is carried out with the ABAQUS [7] finite element package.
FE model establishment
We choose a space frame with an in-plane square
lattice in Figure 1(a) as an example to run the FE
analysis. The initial geometrical assessment has been
carried out in the previous section and the contour plot
of strain energy of the space frame is shown in Figure 4.
In the FE model, the first thing to do is to determine
the three dimensional architecture of the space frame
because so far only the pattern in the top layer and
bottom layer has been considered. In order to remove
the internal mechanism in the structure, some triangular forms have been introduced to connect the top layer
and bottom layer. A single three-dimensional lattice
has been presented in Figure 7 to represent one cell
unit of the space frame. A 12x12 double-layer cylindrical space frame model was then established with all
Initially, truss element T3D2 was used in the analysis. The ABAQUS simulation shows that the space
frame does gradually move to the second configuration under the displacement constraints of 5 nodes.
Figure 8 shows the top view of the initial configuration of the space frame and Figure 9 shows the top
view of the final configuration after the simulation is
completed, corresponding to the second stable state.
It is noted that, in the initial configuration, the axis of
the cylinder on which the mid-plane of the space frame
lies, is direction 2 , while in the second configuration
it has rotated by 90◦ to direction 1. The diagram of
the strain energy variation during the FE analysis is
shown in Figure 10. The energy increases from initially
zero energy to the intermediate high energy level and
goes down to almost zero level at the end of STEP 12.
After that, only 6 rigid body constraints are retained
on the space frame and the energy level decreases to
zero at the end of STEP 13. The ABAQUS results
exactly match the results from the simple geometrical
estimation.
Beam element B31 has also been used to model this
space frame. The analysis procedure is the same as
before. Figure 11 presents the energy change during
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American Institute of Aeronautics and Astronautics Paper 2005-1870
the simulation. Because beam elements can endure
bending deformation in addition to axial deformation,
there is more energy stored in the structure when the
displacement constraints have been applied. Therefore it can be observed in Figure 11 that the energy
level at the second configuration is around half of the
peak value. Also due to the beam elements, it is found
that the ratio of diameter of bars to their length is
crucial for the simulation to converge. Increasing this
ratio may result in the analysis not converging. It is
not reasonable to predict the stability of the second
configuration in this case.
−3
9
2
x 10
8
7
1
Fig. 8
Strain Energy (Nm)
3
Initial stable configuration.
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Step
Fig. 11
ments.
Strain energy variation using beam ele-
Physical models
Research into the constructing of physical models
of these bi-stable space frames has also been carried
out in Cambridge. To achieve monolithic structures,
rapid prototyping technology has been applied to build
these models. Two sample models based on the information gained from ABAQUS analysis have been
selected to be built. The structural parameters and
material properties of model 1 are shown in Table 3.
These parameters are selected to ensure the ABAQUS
simulation can converge when both truss or beam elements are used. The analysis shows that during the
change of the configuration, the maximum von Mises
equivalent stress in the entire model is 20 MPa for
truss elements and 11 MPa for beam elements. Both
cases are lower than the maximum allowed stress 69
MPa.
2
3
1
Fig. 9
Second stable configurations.
−3
8
x 10
7
Strain Energy(Nm)
6
5
Table 3
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Step
Fig. 10
ments.
Parameters of model 1
Initial curvature of mid-plane
Initial rotating angle
Side length of square
Depth of space frame
Radius of cross section of bars
Name of mateial
Tensile modulus of material
Tensile strength of material
Elongation @ break
Strain energy variation using truss ele-
κc0 = 8.3 m−1
θ0 = 45◦
L = 8 mm
d = 6 mm
r = 0.25 mm
SI50 (Resin)
E = 2.48 − 2.69 GPa
48 50 MPa
5.3 − 15.0%
Table 4 presents structural parameters and the material properties of model 2. During the simulation,
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American Institute of Aeronautics and Astronautics Paper 2005-1870
Table 4
Parameters of model 2
Initial curvature of mid-plane
Initial rotating angle
Side length of square
Depth of space frame
Radius of cross section of bars
name of mateial
Tensile modulus of material
Tensile strength of material
Elongation @ break
κc0 = 8.3 m−1
θ0 = 45◦
L = 10 mm
d = 10 mm
r = 0.5 mm
Polyamide PA2200 (Nylon 12)
E = 1.55 − 1.85 Gpa
42 − 48 Mpa
15 − 25%
the maximum von Mises equivalent stress in the model
is 42 MPa for truss elements and 48 MPa for beam
elements. These also below the maximum allowable
stress.
Conclusions and Future work
This paper has described double-layer cylindrical
space frames which have two switchable stable configurations. The bi-stability of the structure was first
analyzed by using a simple geometrical analysis. After that, a more refined Finite Element analysis was
conducted, based on the initial estimation. This analysis has shown that the second configuration is indeed
stable. Additionally, research on constructing physical
models of this kind of bistable space frames has also
been presented. Future work will involve investigation on space frames with other architectures. Further
research must also be carried on to determine more
suitable materials and manufacturing technologies for
the construction of the bi-stable space frames.
Acknowledgments
Financial support from Cambridge-MIT Institute
(CMI), Churchill College and Cambridge University
Engineering Department is gratefully acknowledged.
References
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E. and Pellegrino S., Bi-stable Prestressed Shell
Structures, Cambridge University Engineering Department
Technical Report 2003
2 Iqbal K. and Pellegrino S., Bi-stable composite shells, Proc.
41st AIAAASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 3-6 April 2000, Atlanta GA,
AIAA 2000-1385
3 Schiøler T., Multi-Stablility Structures, Cambridge University PhD thesis 2005
4 Miura K. and Pellegrino S.,Structural Concept, To be published
5 Heki K. and Saka T., Stress analysis of lattice plates as
anisotropic continuum plates, Proc. IASS Pacific Symposium,
1971, Tokyo and Kyoto,Japan, pp663-674. IASS
6 Santer M. and Pellegrino S., Asymmetrically-Bistable
Monolithic Energy-Storing Structures,
Proc. 45th
AIAAASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference Proceedings, 2004
7 ABAQUS, Inc. (2001). ABAQUS Theory and Standard
User’s Manual, Version 6.4, Pawtucket, RI, USA.
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